Find the magnitude and direction of the vector represented by the following pairs of components: (a) Ax = −8.60 cm, Ay = 5.20 cm:

Verified step by step guidance

1. Identify the components of the vector. Here, Ax = -8.60 cm and Ay = 5.20 cm.
2. Calculate the magnitude of the vector using the Pythagorean theorem: \( \text{Magnitude} = \sqrt{Ax^2 + Ay^2} \).
3. Determine the direction of the vector using the tangent function: \( \theta = \tan^{-1}\left(\frac{Ay}{Ax}\right) \).
4. Since Ax is negative and Ay is positive, the vector lies in the second quadrant. Adjust the angle accordingly if necessary.
5. Summarize the magnitude and direction as the final answer.

Key Concepts

Vector Components

Vectors are quantities that have both magnitude and direction. They can be broken down into components along the axes of a coordinate system, typically represented as Ax (horizontal) and Ay (vertical). Understanding how to resolve a vector into its components is essential for analyzing its overall effect in two-dimensional space.

Magnitude of a Vector

The magnitude of a vector is a measure of its length or size, calculated using the Pythagorean theorem. For a vector with components Ax and Ay, the magnitude is given by the formula |A| = √(Ax² + Ay²). This value provides a scalar representation of the vector's strength, independent of its direction.

Direction of a Vector

The direction of a vector indicates where it points in space and is often expressed as an angle relative to a reference axis. For a vector with components Ax and Ay, the direction can be found using the arctangent function: θ = arctan(Ay/Ax). This angle helps in visualizing the vector's orientation in the coordinate system.